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 A120612 For n>1, a(n) = 2*a(n-1) + 15*a(n-2); a(0)=1, a(1)=1. 8
 1, 1, 17, 49, 353, 1441, 8177, 37969, 198593, 966721, 4912337, 24325489, 122336033, 609554401, 3054149297, 15251614609, 76315468673, 381405156481, 1907542343057, 9536162033329, 47685459212513, 238413348924961, 1192108586037617, 5960417405949649 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Characteristic polynomial of matrix M = x^2 - 2x - 15. a(n)/a(n-1) tends to 5, largest eigenvalue of M and a root of the characteristic polynomial. a(2n+1) = A005059(2n+1) = {1,49,1441,37969,966721,...} = (5^(2n+1) - 3^(2n+1))/2. a(2n) = A081186(2n) = {17,353,8177,198593,...} = (3^(2n) + 5^(2n))/2, 4th binomial transform of (1,0,1,0,1,......), A059841. - Alexander Adamchuk, Aug 31 2006 Binomial transform of [1, 0, 16, 0, 256, 0, 4096, 0, 65536, 0, ...]=: powers of 16 (A001025) with interpolated zeros. - Philippe Deléham, Dec 02 2008 a(n) is the number of compositions of n when there are 1 type of 1 and 16 types of other natural numbers. - Milan Janjic, Aug 13 2010 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Index entries for linear recurrences with constant coefficients, signature (2, 15). FORMULA Let M = the 2 X 2 matrix [1,4; 4,1], then a(n) = M^n * [1,0], left term. a(n) = ( 5^n + (-1)^n * 3^n ) / 2. - Alexander Adamchuk, Aug 31 2006 a(n) = Sum_{k, 0<=k<=n}A098158(n,k)*16^(n-k). - Philippe Deléham, Dec 26 2007 If p=1, and p[i]=16, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010 EXAMPLE a(4) = 353 = 2*49 + 15*17 = 2*a(3) + 15*a(2). MATHEMATICA Table[(5^n+(-1)^n*3^n)/2, {n, 1, 30}] - Alexander Adamchuk, Aug 31 2006 a[n_] := (5^n + (-3)^n)/2; Array[a, 24, 0] (* Or *) CoefficientList[Series[(1 + 15 x)/(1 - 2 x - 15 x^2), {x, 0, 23}], x] (* Or *) LinearRecurrence[{2, 15}, {1, 1}, 25] (* Or *) Table[ MatrixPower[{{1, 2}, {8, 1}}, n][[1, 1]], {n, 0, 30}]  (* Robert G. Wilson v, Sep 18 2013 *) PROG (PARI) a(n)=([1, 4; 4, 1]^n)[1, 1] \\ Charles R Greathouse IV, Oct 16 2013 (PARI) concat(1, Vec((15*x+1)/(-15*x^2-2*x+1) + O(x^100))) \\ Colin Barker, Mar 12 2014 (PARI) a(n) = ( 5^n + (-1)^n * 3^n ) / 2 \\ Charles R Greathouse IV, May 18 2015 CROSSREFS Cf. A005059, A081186, A059841. Sequence in context: A146831 A146698 A146706 * A146461 A098329 A160076 Adjacent sequences:  A120609 A120610 A120611 * A120613 A120614 A120615 KEYWORD nonn,easy AUTHOR Gary W. Adamson, Jun 17 2006 EXTENSIONS More terms from Alexander Adamchuk, Aug 31 2006 Entry revised by Philippe Deléham, Dec 02 2008 More terms from Colin Barker, Mar 12 2014 STATUS approved

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Last modified September 19 12:57 EDT 2019. Contains 327198 sequences. (Running on oeis4.)